M. Nikolić i dr. Pregled numeričkih metoda za modeliranje u mehanici stijena

ISSN 1330-3651 (Print), ISSN 1848-6339 (Online) DOI: 10.17559/TV-20140521084228

OVERVIEW OF THE NUMERICAL METHODS FOR THE MODELLING OF ROCK MECHANICS PROBLEMS

Mijo Nikolić, Tanja Roje-Bonacci, Adnan Ibrahimbegović

Subject review The numerical methods have their origin in the early 1960s and even at that time it was noted that numerical methods can be successfully applied in various engineering and scientific fields, including the rock mechanics. Moreover, the rapid development of computers was a necessary background for solving computationally more demanding problems and the development process of the methods in general. Thus, we have many different methods presently, which can be separated into two main branches: continuum and discontinuum-based numerical methods. Some problems require the strengths of both main approaches which brought the hybrid continuum/discontinuum methods. The first goal of this paper is to present the state of the art numerical methods and approaches for solving the rock mechanics problems, as well as to give the brief explanation about the theoretical background of each method. The second goal is to emphasise the area of applicability of the methods in rock mechanics.

Keywords: numerical methods; numerical modelling; rock mechanics

Pregled numeričkih metoda za modeliranje u mehanici stijena

Pregledni rad Počeci numeričkih metoda sežu u rane 1960-e. Već tada je bilo jasno da numeričke metode mogu biti uspješno upotrijebljene za različita inženjerska i znanstvena područja, uključujući i primjenu u mehanici stijena. Ubrzan razvoj računala je omogućavao razvoj numeričkih metoda i rješavanje računalno zahtjevnijih sustava. Takav razvoj doveo je do velikog broja različitih metoda i pristupa koji se mogu svrstati u dvije skupine: metode kontinuuma i metode diskontinuuma. Određene zadaće zahtijevaju prednosti oba pristupa, što je dovelo do razvoja kombiniranih konačno-diskretnih metoda. Prvi cilj ovog rada je predstavljanje numeričkih metoda i pristupa koji se koriste za rješavanje zadaća u mehanici stijena, kao i kratko objašnjenje osnovnih teorijskih postavki svake od metoda. Drugi cilj je osvrt na primjenjivost pojedine metode u mehanici stijena.

Ključne riječi: numeričke metode; numeričko modeliranje; mehanika stijena

1 Introduction development of computers made a significant contribution to the field called computational mechanics which found Rock material is a natural geological material its wide use in rock mechanics. Namely, the numerical consisting of different minerals. It is discontinuous, methods for approximately solving the partial differential anisotropic, inhomogeneous, inelastic and contains equations are used today as the main approach in numerous randomly oriented zones of initiation of engineering design and research of rock mechanics. In potential failure. These zones can be initial cracks, this paper we are presenting the state of the art of defects, cavities or other natural flaws. Rocks have been numerical methods in terms of historical background as used (by humans) from the early beginnings of the human well as their strengths and weaknesses with the focus on race in many different ways especially for the building deterministic approach. purposes. We were constantly facing its unpredictable Numerical model for the analysis of rock mass nature trying to build roads, tunnels, underground problems has to include the possibility of description of excavations, mining shafts or avoid the sudden rock falls the rock behaviour by relevant material model and and slopes. From the early days there was a need for fracture mechanism, the presence of the pre-existing predicting the behaviour of this material and up to present cracks, the pre-existing stress state, inhomogeneity and times, we are still trying to fully understand it. anisotropy as well as a time dependent behaviour caused Because of the irregular nature of rock material, the by creep and plastic deformation. The additional prediction of its behaviour has always been a challenging challenge is coupling of the hydraulic and thermal process task. Rock mechanics developed mostly for the design of in analysing the mechanical behaviour of the rock. The rock engineering structures and today we have a wide choice of the appropriate model and numerical methods spectrum of different modelling approaches for many depends on the problem which has to be solved. various rock mechanics problems. That includes the The numerical methods can be classified into three approaches based on the previous experiences, simplified main categories: continuum, discontinuum and hybrid mathematical models that can be solved analytically (e.g. continuum/discontinuum methods. Namely, the Bishop Method of slices for slope stability) or general continuum concept implies that the domain of interest classification systems for rocks. The latter found large cannot be separated and the continuity between the points number of applications in various types of engineering must be preserved in order to establish the derivatives. projects like designing and construction of excavations in Thus the continuity between the elements in numerical rock. There are several rock mass classification systems approach must be preserved as well. Contrary to the including RMR, Q and GSI classifications which are used analytical solution of differential equations where the to estimate the rock mass properties. The lack of solution is known in each point, the numerical solution is information is a common fact in rock mechanics and calculated in the finite number of pre-defined nodes engineering design so empirical approaches (including the which reduces the system complexity. The discontinuum classification systems) are also widely used. The approach on the other side treats the separate elements as

Tehnički vjesnik 23, 2(2016), 627-637 627 Overview of the numerical methods for the modelling of rock mechanics problems M. Nikolić et al. discrete ones which are individually continuous and they at the surrounding nodes. After introducing the boundary interact between each other. In the discontinuous conditions, the system of algebraic equations is finally methods, the rigid body motion (usually with large solved (usually by direct or iterative methods) which movements) is the main case of interest, while in produces the values of unknowns at each node leading to continuum-based methods the deformation of the system the approximate solution of the partial differential is the main objective. Thus, the sliding of the rock block equation with an error made because of the difference in on the pre-existing discontinuity would rather be partial derivatives and finite differences. Such a direct computed with the discrete methods, while the choice of kind of discretization, together with no practical need to continuum method would be better used in calculating the use interpolation functions as in other methods like FEM deformation of the rock mass above the excavation of the or BEM, position this method as the most direct and tunnel. In addition there also exist the hybrid intuitive technique for solving the partial differential continuum/discontinuum methods that use the best equations. The advantage of the method is also in properties of both approaches. Such an example is to have possibility of the simulation of complex nonlinear the discrete rock block which moves and also deforms material behaviour without iterative solutions. under the external loading. The standard FDM uses regular grid, such as rectangular which is also the most important shortcoming Table 1 Overview of the most notable numerical approaches and of the method. Thus, representing the irregular geometries methods together with dealing with heterogeneous nature of rocks Continuum Methods: and complex boundary conditions seems like a significant ▪Finite Difference Method FDM ▪Finite Volume Method FVM disadvantage of this method for simulating rock ▪Finite Element Method FEM behaviour in practical rock mechanics tasks. Another ▪Meshless Methods important disadvantage lies in continuity requirement of ▪Boundary Element Methods BEM the proposed equations which is not suitable for dealing Discontinuum Methods: with fractures. However, the general FDM method [1, 2] ▪Discrete Element Method DEM evolved eventually to deal with these shortcomings, ▪Discrete Fracture Network Method DFN Hybrid Methods: mostly with irregular grids, which include irregular ▪Discrete Finite Element Method quadrilateral, triangular and Voronoi grids. ▪Combined Finite Discrete Element Method FEM/DEM Further development of this approach continued with the Finite Volume Method (FVM) [3]. The FVM is also a The most notable continuum numerical methods are method for solving partial differential equations, not Finite Difference Method (FDM), Finite Volume Method directly as in FDM, but in integral sense. This leads to the (FVM), Finite Element Method (FEM), Meshless formulation of finite volumes, which represent the volume Methods and Boundary Element Method (BEM). The surrounding each node in mesh. The basic principle is to discontinuum methods are Discrete Element Method replace the integrals with algebraic functions of the (DEM) and Discrete Fracture Network (DFN). The most unknowns in the nodes, taking into account the initial and notablehybrid continuum/discontinuum methods with boundary conditions which lead to the set of algebraic applications in rock mechanics are Discrete Finite equations. Element Method, Combined Finite Discrete Element The main advantages of the FVM are the possibilities Method (Tab. 1). of using the irregular unstructured meshes such as triangles, arbitrary quadrilaterals or Voronoi cells and 2 Overview of the numerical methods considering the material heterogeneities [4]. Namely, it is 2.1 The finite difference and finite volume method possible to join different material properties to different (FDM, FVM) finite volumes. The continuity requirement between the neighbouring The Finite Difference Method (FDM) is one of the nodes still makes the fracture propagation impossible to oldest widely-applied numerical methods for solving the include, which is a main disadvantage of FDM/FVM. The partial differential equations that found their application analysis of the fracturing processes in FDM/FVM models in the rock mechanics. The general principle of the can be conducted through the material failure at the nodes method is replacing the partial derivatives of the function or cell centres, but in this way it is not possible to by the finite differences defined over a certain interval in simulate the true fracture propagation. Despite this the coordinate directions. More precisely, the domain lacking, the FVM is still one of the most popular needs to be partitioned into grid of nodes, among which numerical methods in rock mechanics with applications in the finite differences are defined. The different choices slope stability, tectonic process, rock mass can be made for finite difference integration schemes: characterization and especially in coupled hydro- explicit or forward, implicit or backward and Crank- mechanical problems. The overview of the FVM Nicolson or central difference scheme are largely used. It applications in rock mechanics can be seen in [5]. The is worth noting the FDM can be used for solving the time latest improvements of the FDM/FVM can be seen in [6- dependent problems. 9]. One of the most well-known computer programs for As a result of meshing the domain, a system of solving a variety of rock mechanics problems by algebraic equations with unknowns related to the pre- FDM/FVM is FLAC/FLAC3D [10]. defined nodes will arise. Each algebraic equation connected to its corresponding grid node is expressed as combination of function values at its own node, as well as

628 Technical Gazette 23, 2(2016), 627-637 M. Nikolić i dr. Pregled numeričkih metoda za modeliranje u mehanici stijena

2.2 The finite element method (FEM) discontinuities with the 'joint' elements where their influence on physical behaviour is considered through The Finite Element Method (FEM) originates from constitutive laws of the discontinuities as equivalent the early 1960s [11, 12]. It was developed as an continuum. However, no real detachment is possible with alternative to the finite difference method for the them. The 'joint' elements are also limited to small numerical solution of stress concentration in continuum displacements, while large movements across the mechanics and was the first numerical method which was discontinuities (e.g. sliding on the discontinuities) are not able to account for material heterogeneities, non- possible due to the continuum assumptions. The first linearities, complex geometries and boundary conditions. element of this kind was 'Goodman joint element' Due to this, FEM immediately became the most applied developed especially for rock applications [24, 25]. Later numerical method in rock mechanics, especially because improvements of the joint elements can be found in the FDM at that time was limited only to regular grids. literature [26÷29]. The rapid application of the method started in the late The ultimate load computation where structural 1970s and early 1980s with the early works of elements are subjected to progressive failure which leads Zienkiewicz and Bathe [13, 14]. Following these works, to the collapse of the structure has been the topic of many rock mechanics problems at that time were solved interest of many authors recently since this is one of the with FEM [15÷17]. The FEM was developing further crucial types of failure mechanisms. The key difficulty in during the years, and today it is still the most applied failure analysis is correct and mesh-independent numerical method for many advanced rock and soil representation of the post-peak softening behaviour mechanics simulations of the non-linear, time-dependent related to crack propagation. Also, when trying to and anisotropic behaviour. simulate the fracture growth with FEM, it is essential to The FEM is the numerical method for finding have small size of elements and perform the continuous approximate solutions to the boundary value problems for re-meshing as the crack propagates. The 'enhanced' FEM differential equations. The general principle is to divide methods have been evolving to overcome these the domain of the problem into smaller sub-domains difficulties, and a couple of new methods derived to help called finite elements, do the local approximation inside alleviate shortcomings of the standard FEM. On one side each finite element, perform the finite element assembly there is the finite element method with embedded and find the solution of the global matrix equation. More discontinuities (ED-FEM), representing cracks truly in precisely, the unknown function (usually displacement each element (e.g. see [30÷38]). On the other side there is function) needs to be approximated with a trial function extended finite element method (X-FEM) where cracks of the nodal values in a polynomial form, where the are represented globally (e.g. [39÷42]). The ED-FEM and numerical integration is performed in each element in X-FEM methods are equivalent in their capabilities to Gaussian quadrature points. After the finite element handle the most demanding kinematics incorporating both assembly is performed, the algebraic system of equations strong and weak discontinuities (e.g. see [43]). The strong on a global level is obtained. The FEM derived as a discontinuities serve for truly simulating the crack special case of Galerkin method where trial functions are propagation, while the weak discontinuities enable the presented globally, contrary to local approximation in heterogeneous representation of material within element. FEM. General approach is to enhance the standard kinematics of One of the mostly used advantages of FEM is a the finite elements with additional discontinuous possibility of representing heterogeneous rocks, where it functions to simulate discontinuous behaviour. Thus, the is possible to assign different material properties to enhanced kinematics in the strain and displacement fields different finite elements. Presently, there are many serves respectively for dealing with heterogeneities and various shapes of finite elements with different number of localization phenomena. The ED-FEM relies upon the nodes for 1D, 2D and 3D cases. Special case of elements incompatible mode method and Hu-Washizu mixed called the 'infinite elements' was developed to simulate variational formulation [44, 45]. The most recent the far-field domain in geotechnical applications [18÷20]. contribution to ED-FEM approach in rock mechanics was Because of the continuum assumptions, the usual made in the following works [46÷49] where the crack FEM methods have restrictions in efficient application of propagation due to modes I, II and III, as well as their the failure analysis, cracking and damage induced combination, provides successful simulation of complex discontinuities or singularities [21, 22]. Since rock is a failure mechanisms that occur in rocks. This kind of discontinuous material and FEM is a continuum method, representation is also suitable for connecting the scales there have been many attempts to improve it in order to where the natural crack growth from the micro to macro simulate the fracture propagation and other discontinuous scale can be simulated. Namely, the cracks form as a effects with it. From the early experiments on rock result of accumulation of micro-cracks leading to forming specimens, it was observed that the experimentally larger macro-cracks. obtained stress-strain curves up to failure are not linear. Another example of enhanced finite elements is a One of the earliest models that could approximately Generalized Finite Element Method (G-FEM) [50, 51]. simulate the stress-strain nonlinear curve due to crack The G-FEM uses the local functions which are usually opening was smeared-crack models [23]. These models analytical solutions to specific problems. These additional were perfectly brittle in the beginning, although the rock functions are not necessarily polynomials, but are bonded has some residual load-carrying capacity after reaching its together with the standard space through the partition of strength resulting with softening behaviour. Another unity principle [52]. The advantage of G-FEM is that attempt was an indirect representation of the complex geometries can be represented easily because the

Tehnički vjesnik 23, 2(2016), 627-637 629 Overview of the numerical methods for the modelling of rock mechanics problems M. Nikolić et al. mesh is independent of the geometry of the domain of the 2.4 The boundary element method (BEM) problem. The fractures can be simulated by additional functions and their corresponding additional nodes The BEM is a numerical computational method for surrounding the crack. The other strength of this approach solving partial differential equations where the solution of is the possibility to represent voids, problems with micro- weak form is obtained globally through an integral scales and boundary layers, which is important for rock statement. The basic principle of BEM is using the given mechanics engineering. boundary conditions to fit boundary values into the Apart from FEM being suitable for representing integral equation. Thus the discretization is needed only at heterogeneous materials and using the unstructured and the boundary with a finite number of boundary elements. irregular meshes, it also proved to be the appropriate tool After finding the solution on a boundary, the integral for representing various non-linear and inelastic types of equation can be used again for calculation of solutions behaviour. These observations carried the challenges up directly inside the domain. The main advantage of the to present time and many different models have been BEM is reduction of model dimensions by 1, so for 2D made for representation of material hardening and BEM problems, the boundary elements are 1D lines that softening. Among them damage [53, 54] and plasticity can be constant, linear or quadratic. In the 3D case, the rock models [55÷57] are the most used frameworks for boundary elements are 2D elements. The approximation simulation of the non-linear and inelastic behaviour of of the solution at the boundary elements is performed material. The FEM is also suitable for representing the using the shape functions similarly to FEM. The geometric non-linearities, contact mechanisms, fluid- numerical integration is usually performed by using structure interaction, connecting the scales from nano and Gaussian quadrature points. When applying the boundary micro scale to large macro scales, etc. All of these reasons conditions into the matrix equations obtained after the position the FEM as the mostly used numerical method in approximation stage, the final global matrix equation with rock mechanics. The generalized and enhanced FEM unknowns at boundary is obtained. Contrary to standard makes the method even more attractive because of its FEM matrix equations, the global matrix in BEM is capabilities of fracture simulations without re-meshing. usually asymmetric. One of the most notable commercial FEM software for geotechnical applications including rock mechanics is Table 2 The most representative and widely used methods in the PLAXIS [58]. continuum approach ▪Standard Finite Difference Method FDM 2.3 The meshless methods ▪General Finite Difference Method FVM FDM/ ▪Finite Volume Method FVM

The application of the FEM in practical engineering ▪ Standard Finite Element Method problems with complex geometries, material properties ▪Extended Finite Element Method X-FEM ▪Generalized Finite Element Method G-FEM FEM and boundary conditions requires mesh generation which ▪Embedded Discontinuity Finite Element Method ED-FEM sometimes, especially in 3D problems, can be equally as ▪ Smoothed particle hydrodynamics demanding as solving the problem. Another weakness of ▪ Diffuse element method the FEM is locking phenomenon which is often reflected ▪Element-free Galerkin method in a numerical instability due to the mesh distortion. Both ▪Reproducing kernel particle methods of these problems can be avoided by meshless methods ▪Moving least-squares reproducing kernel method

Meshless ▪Hp-cloud method which developed in a way that elements connecting the ▪ Method of finite spheres nodes are not required. Instead, the trial functions are no ▪Finite point method

longer as in standard FEM, but generated from the ▪ Dual Boundary Element Method (dual BEM) neighbouring nodes within a domain of influence. More ▪Galerkin BEM precisely, it is only necessary to generate the nodes across BEM ▪Dual Reciprocity BEM the domain without defining fixed element topology. The interpolation functions obtained in this way are no longer The BEM application in rock mechanics has its polynomial functions and it is more difficult to integrate origins in the early 1980s [68, 69]. This method found them compared to standard FEM where Gauss integration applications in underground excavations [70÷72], points inside the elements are used for numerical dynamic rock problems [73], analysis of in situ stress [74] integration. This makes the meshless methods and borehole drilling [75]. Besides reducing the model computationally more demanding, but with an advantage dimensions by 1, its strength is accuracy in finding the of not using the standard meshes generators and easier solution because of its direct integral formulation. The representation of more complex geometries. Many standard BEM was developed for continuous and linear meshless methods have been developed, but some of the elastic solutions inside domain which was a disadvantage mostly used ones are: Smoothed particle hydrodynamics of the method in the beginning. The other main [59], Diffuse element method [60], Element-free Galerkin disadvantage over FEM is not efficient dealing with method [61], Reproducing kernel particle methods [62], heterogeneity because not complete domain is discretized Moving least-squares reproducing kernel method [63], with elements. The fracture propagation with BEM was hp-cloud method [64], the method of finite spheres [65], possible with the later development of two new Finite point method [66] etc. For the specific purpose in approaches. The first one is using the domain division rock mechanics, where the treatment of the fractures is of into sub-domains and the pre-defined crack path [76]. The essential influence, the meshless approach has significant second one was Dual BEM (DBEM) [77] with application potential which is shown by Zhang et al [67].

630 Technical Gazette 23, 2(2016), 627-637 M. Nikolić i dr. Pregled numeričkih metoda za modeliranje u mehanici stijena of the displacement and traction boundary conditions at problems in rock mechanics using distinct element opposite side of the fracture. With these improvements, method are UDEC and 3DEC [102]. the BEM became a notable tool for simulating the The most well-known method of implicit DEM fracturing process. The other variants of BEM are approach is Discontinuous Deformation Analysis (DDA) Galerkin BEM (GBEM) [78, 79] and Dual-reciprocity [103, 104]. It is similar to the finite element method for BEM (DRBEM) [80, 81]. The list of most notable solving deformation of the bodies, but it also accounts for continuum methods is given in Tab. 2. interaction of the independent blocks along discontinuities in fractured and jointed rock mass. DDA uses an implicit 2.5 The discrete element method (DEM) algorithm for simultaneous solution of the equations of equilibrium by minimizing the total potential energy of The DEM started to develop in the field of the rock the blocky rock mass system. The interaction between the mechanic applications due to its requirements for independent blocks is performed by equations of contact modelling the discontinuous behaviour [82, 83]. The interpenetration and accounts for friction. DDA uses the method was primarily defined as the computational large displacement formulations for discontinuous approach that can simulate finite displacements and movements between blocks. In this method, the standard rotations of discrete bodies including their detachment. mesh generation is used similarly to FEM, while penalty The theoretical formulations are based upon the equations method or Lagrange multiplier method are used for the of motion of rigid or deformable bodies using implicit or contact. The early formulations were limited to only explicit time integrations. The basic concept is to treat the simple representation of regular block movements and domain of interest as an assembly of particles or blocks deformations, while in the later developments irregular which are continuously interacting between each other. In blocks could be used and meshed with the triangular and the DEM approach, the contact between components of four-node finite elements [105, 106]. It is an attractive the system is constantly changing during the deformation method for solving rock mechanics problems where the process. rock blocks interact and deform between each other. The Later on, many different numerical methods based on list of applications is covered in [107÷109]. DEM developed for various rock mechanics problems. Besides using the block systems in rock mechanics, The main strength of the approach was the fact that the the particle based DEM is an appropriate method for real discontinuities could be simulated, as well as simulating the granular materials such as soil. The representing the rock blocks which move and interact principle of solving tasks with granular materials is the between each other including the fragmentation process same as for blocks, but with the simpler contact etc. All of DEM based methods have the similar basic mechanisms due to rigid regular or irregular particles that approach, while the differences between them are the use do not deform. The regular particles include circular, of various shapes of discrete elements, the way of elliptical and ellipsoidal shapes, while irregular ones can computing the contact forces between the discrete bodies, be polygonal or polyhedral. Some of the applications of the way of recognizing the contact, the way of integration particle based DEM in rock mechanics are rock fracturing of equations of motion, etc. The contact between the and fragmentation due to rock blasting [110÷112], discrete bodies is essential part of solving the task. tunnelling [113], hydraulic fracturing [114] and ground Nowadays, the mostly used contact algorithms are movements [115]. penalty, Lagrangian multiplier and augmented Lagrangian Another approach similar to DEM is the Dynamic types of contact. The overview of the contact models can lattice network method used to simulate rock fracture be found in [84]. initiation and propagation. The general idea is to have According to the numerical integration scheme, there rigid particles covering the domain of interest, with exist the two main approaches of DEM: explicit and springs acting as cohesive links and connecting the implicit time integration DEM. The application of the particles. The springs usually do not possess the mass and explicit DEM started in the 1970s and was used mostly have the characteristic stiffness properties of material. for rigid block simulations. The most representative The mass of the particles depend on the density of the explicit DEM method is Distinct Element Method material. During the simulation process, the springs developed by Cundall [85, 86]. It was created to simulate deform until they are completely broken leaving the the fracturing, cracking and splitting of the blocks under particles to move and interact with other particles. The external loading. The method was applied in many application examples in rock engineering are mostly used various rock mechanic applications including tunnelling, in fracturing processes of intact rock [116÷118]. One underground works and rock dynamics [87÷91], nuclear example of lattice models is found in [119], where waste [92, 93], rock slopes [94], acoustic emission in rock particles are represented with Voronoi cells with [95], boreholes [96], laboratory test simulations [97], etc. geometrically nonlinear Reissner beams acting as It is worth noting that irregular geometry of rocks is cohesive links between them. Such a Voronoi cell usually not easy to incorporate into numerical representation allows for simulating the fracturing of the simulations. Thus, one of the latest and modern brittle heterogeneous material such as rock and concrete. approaches to capture the realistic geometry and use it Lately, many numerical methods that combine with DEM software is LIDAR laser scanning technology advantages of finite and discrete elements developed. [98, 99]. The overview of explicit DEM approaches in One of the well-known methods which combine finite and rock engineering can be found in [100, 101]. The most discrete elements is Discrete Finite Element Method in notable computer codes for two and three-dimensional which the four-node blocks are used [120÷122]. The main advantage of the method is a possibility to simulate

Tehnički vjesnik 23, 2(2016), 627-637 631 Overview of the numerical methods for the modelling of rock mechanics problems M. Nikolić et al. interaction between the blocks, as well as the deformation irregular rock properties such as heterogeneity, as well as of blocks. The higher order shape functions are used here the fracturing process, were some of the main difficulties for considering non-linear deforming of the blocks. The that influence the choice of method. The FDM is widely method uses implicit time integration and it is similar to used because of its simplicity and the possibility of DDA. Another similar approach is Combined Finite handling the non-linear behaviour. At its beginnings, Discrete Element Method (FEM/DEM) [123, 124] which FDM was limited to regular mesh and thus was not considers block deformation as well as the fragmentation capable of simulating irregular geometries and complex processes. Contrary to Discrete Finite Element Method, boundary conditions. Thanks to the more recent the FEM/DEM uses explicit time integration scheme. The developments, the general FDM overcame these FEM/DEM method is used in rock mechanics difficulties and is presently capable to model complex and applications [125, 126] and simulation of the behaviour of irregular geometries. The FVM was capable to represent stone structures [127]. There were some developments of complex geometries and material heterogeneities, but due the method considering the reinforcement between to representing the material failure at the nodes or cell discrete elements with application in concrete, but which centres, it is not possible to create fracture propagation. can be easily used to simulate the rock slopes or tunnel The FEM has been mostly used among all of the excavations with anchor reinforcements [128]. The list of numerical methods since its beginnings, because of its the covered discontinuum methods is given in Tab. 3. capabilities to represent material heterogeneities, non- linear behaviour such as damage and plasticity, complex Table 3 The most representative and widely used methods in the geometries and boundary conditions. The original FEM discontinuum approach could simulate the propagation of discontinuities using

▪Discontinuous Deformation Analysis DDA (implicit) the re-meshing process which is computationally ▪Distinct Element Method (explicit) demanding especially for 3D cases. The constant ▪Particle Discrete Element Methods DEM ▪Dynamic Lattice Network Methods development led to the enhanced FEM methods such as X-FEM, ED-FEM, G-FEM and meshless methods which The flow equations are solved with: ▪Closed form solutions enable to represent the true crack propagation and other ▪FEM localized and discontinuous effects which happen

DFN ▪BEM regularly in the rock mass. The main strength of BEM is ▪Pipe model to represent fracturing in rocks due to the most recent ▪Lattice models formulations, to reduce the problem complexity from 3D

to 2D, or 2D to 1D and solve the problem at boundary. 2.5 Discrete fracture network method (DFN) This is appropriate in solving the large scale problems,

where number of degrees of freedom drastically The DFN is a discrete model which started to develop decreases. The DEM approach solves the equations of in the late 1970s mostly for simulating fluid flow and motion and allows de-bonding and detaching of elements, transport processes through the connected rock fractures. thus representing real discontinuities. This is suitable for The groundwater transport is dominated by the connected problems with large number of fractures which are paths formed by fractures, cracks, voids, etc. Thus the dominant in failure process. This method was primarily central problem is to define the position, geometry and developed for the rock mechanics problems and presently properties of discrete fractures. Once the discrete fractures it is widely used for many rock applications. are defined deterministically or stochastically, the DFN It is shown in this paper that presently many different model can be solved to understand the transport issues methods and approaches exist. There is no generalized and flow nature. The flow equations within the fractures method which solves all of the rock mechanics problems, can be performed using closed-form solutions [129], FEM but rather we need to choose the method which is most [130], BEM [131], pipe model [132] or the lattice model appropriate for the problem being solved. [133]. For the smooth and regular fractures close-form solutions can be used, while for the geometrically Acknowledgements irregular shapes, discretization needs to be performed and solution is found numerically. The pipe models and This work was supported by a scholarship from the channel lattice models are solved analytically and are French Government and Campus France. This support is computationally less demanding than FEM and BEM gratefully acknowledged. models. Most notable software incorporating DFN is

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Authors’ addresses

Mijo Nikolić, doctorant 1École Normale Supérieure de Cachan 61 Avenue du Président Wilson, 94230 Cachan, France 2Faculty of Civil Engineering, Architecture and Geodesy, University of Split Matice Hrvatske 15, 21000 Split, Croatia [email protected]

Tanja Roje-Bonacci, full prof. Faculty of Civil Engineering, Architecture and Geodesy, University of Split Matice Hrvatske 15, 21000 Split, Croatia [email protected]

Adnan Ibrahimbegović, prof. classe exceptionnelle Chair for Computational Mechanics UT Compiegne/Sorbonne Universités Lab. Mécanique Roberval Centre de Recherches Royallieu, 60200 Compiegne, France [email protected]

Tehnički vjesnik 23, 2(2016), 627-637 637